the scalar field dependent tensor ΦABΛ (φ) being intrinsically defined as the coefficient of the term εAψμB in the supersymmetry transformation rule of the vector field AΛμ , namely:

From its own definition it follows that under isometries of the scalar manifold ΦABΛ (φ) must transform in the representation R of Giso times >2 N of the R- symmetry USp(N ). In the case of N = 8 supergravity the tensor ΦABΛ (φ) is simply the inverse coset representative:

We see in the next subsection how the same object is generally realized in an N = 2 theory via very special geometry.

8.6.1 Very Special Geometry

Very special geometry is the peculiar metric and associated Riemannian structure that can be constructed on a very special manifold. By definition a very special manifold V S n is a real manifold of dimension n that can be represented as the following algebraic locus in Rn+1:

is a constant symmetric tensor fulfilling some additional defining properties that we will recall later on.

The very special metric on the very special manifold is nothing else but the pullback on the algebraic surface (8.6.20) of the following Rn+1 metric:

In other words in any coordinate frame the coefficients of the very special metric are the following ones:

where we have introduced the new objects:

and introduce the 2(n + 1)-vectors:

taking a second covariant derivative it can be shown that the following identity is true:

where the world-index symmetric coordinate dependent tensor Tij k is related to the constant tensor dΛΓ Σ by:

The identity (8.6.29) is the real counterpart of a completely similar identity that holds true in special Kähler geometry and also defines a symmetric 3-index tensor. In the use of very special geometry to construct a supersymmetric field theory the essential property is the existence of the section XΛ(φ). Indeed it is this object that allows the writing of the tensor ΦABΛ (φ) appearing in the vector transformation rule (8.6.18). It suffice to set:

Why do we call it a section? Since it is just a section of a flat vector bundle of rank n + 1

with base manifold the very special manifold and structural group some subgroup of the n + 1 dimensional linear group: Giso GL(n + 1, R). The bundle is flat because the transition functions from one local trivialization to another one are constant matrices:

g Giso : XΛ(gφ) = M[g] ΛΣ XΣ (φ); M[g] = constant matrix (8.6.33)

The structural group Giso is implicitly defined as the set of matrices M that leave the dΛΓ Σ tensor invariant:

Since the very special metric is defined by (8.6.25) it immediately follows that Giso is also the isometry group of such a metric, its action in any coordinate patch (8.6.22) being defined by the action (8.6.33) on the section XΛ. This fact explains the name given to this group.

By means of this reasoning we have shown that the classification of very special manifolds is fully reduced to the classification of the constant tensors dΛΓ Σ such that their group of invariances acts transitively on the manifold S V n defined by (8.6.20) and that the special metric (8.6.25) is positive definite. This is the algebraic problem that was completely solved by de Wit and Van Proeyen in [11]. They found all such tensors and the corresponding manifolds. There is a large subclass of very special manifolds that are homogeneous spaces but there are also infinite families of manifolds that are not G/H cosets.

8.6.2The Very Special Geometry

of the SO(1, 1) × SO(1, n)/SO(n) Manifold

As an example of very special manifold we consider the following class of homogeneous spaces:

SO(1, n)

(8.6.35)

SO(n)

This example is particularly simple and relevant to string theory since reducing it on a circle S1 from five to four dimensions one finds a supergravity model where the special Kähler geometry is that of

which constitutes a primary example with very large applications.

To see that the RT [n] are indeed very special manifolds we consider the following instance of cubic norm:

It is immediately verified that the infinitesimal linear transformations XΛ → XΛ + δXΛ that leave the cubic polynomial C(X) invariant are the following ones:

The transformation δ generates an SO(1, 1) group that commutes with the SO(1, r + 1) group generated by the transformations δL, δu, δv and δA, hence the symmetry group of the symmetric tensor:

d0+− = 1

0 otherwise

defined by the cubic polynomial C(X) is indeed the group SO(1, 1) × SO(1, n). This is quite simple and evident. What is important is that the same group has also a transitive action on the manifold defined by the equation C(X) = 1 that can be identified with the product SO(1, 1) × SO(1, n)/SO(2). To verify this statement it suffices to consider that the quadratic equation

defines the homogeneous manifold SO(1, n)/SO(2) on which SO(1, n) has a transitive action. For instance we can use as independent r + 1 coordinates the following ones:

and then it suffices to set: